Problem: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$3.50$ each for students, and the group paid $$44.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$19.50$ each for teachers and $$8.00$ each for students, and the group paid $$111.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+3.5y = 44.5}$ ${19.5x+8y = 111}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-19.5x-10.5y = -133.5}$ ${19.5x+8y = 111}$ Add the top and bottom equations together. $ -2.5y = -22.5 $ $ y = \dfrac{-22.5}{-2.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {6.5x+3.5y = 44.5}$ to find $x$ ${6.5x + 3.5}{(9)}{= 44.5}$ $6.5x+31.5 = 44.5$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {19.5x+8y = 111}$ and get the same answer for $x$ ${19.5x + 8}{(9)}{= 111}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.